A parabolic PDE-based approach to Borell–Brascamp–Lieb inequality

Abstract In this paper, we provide a new PDE proof for the celebrated Borell–Brascamp–Lieb inequality. Our approach exploits a deep connection between the Borell–Brascamp–Lieb inequality and properties of diffusion equations of porous medium type, especially pertaining to the large time asymptotics and to the preservation of a generalized concavity of the solutions. We also develop this PDE method to recover the equality condition of the Prékopa–Leindler inequality by further exploiting known properties of the heat equation, including the eventual log-concavity and backward uniqueness of solutions.